Vol. 89, No. 1, 1980

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Dense strong continuity of pointwise continuous mappings

P. S. Kenderov

Vol. 89 (1980), No. 1, 111–130

Let Y be a topological space and Z be a metric space with metric d(,). Denote by C(Y,Z) the space of all continuous functions from Y into Z. For a given topological space X and a point wise continuous mapping T : X C(Y,Z) a theorem is proved asserting (under some conditions) that T is continuous at the points of some dense Gδ subset of X with respect to the topology of uniform convergence in C(Y,Z). A “set-valued” version of this result is also proved. It is shown how one can use these results in order to get new information about points of continuity and single-valuedness of (multivalued) monotone operators and (multivalued) metric projections. As corollaries some known results about Gâteaux or Fréchet differentiability of convex functions on a dense subset of their domains of continuity are obtained.

Mathematical Subject Classification 2000
Primary: 46B22
Secondary: 47H99, 54C60
Received: 7 March 1979
Revised: 24 July 1979
Published: 1 July 1980
P. S. Kenderov